Past

Abstract: Sharp asymptotic lower bounds of the expected quadratic

variation of discretization error in stochastic integration are given.

The theory relies on inequalities for the kurtosis and skewness of a

general random variable which are themselves seemingly new.

Asymptotically efficient schemes which attain the lower bounds are

constructed explicitly. The result is directly applicable to practical

hedging problem in mathematical finance; it gives an asymptotically

optimal way to choose rebalancing dates and portofolios with respect

to transaction costs. The asymptotically efficient strategies in fact

reflect the structure of transaction costs. In particular a specific

biased rebalancing scheme is shown to be superior to unbiased schemes

if transaction costs follow a convex model. The problem is discussed

also in terms of the exponential utility maximization.

Turbidity currents - submarine flows of sediment - are capable of transporting particulate material over large distance. However direct observations of them are extremely rare and much is inferred from the deposits they leave behind, even though the characteristics of their source are often not known. The submarine flows of volcanic ash from the Soufriere Hills Volcano, Monsterrat provide a unique opportunity to study a particle-driven flow and the deposit it forms, because the details of the source are relatively well constrained and through ocean drilling, the deposit is well sampled.

We have formed simple mathematical models of this motion that capture ash transport and deposit. Our description brings out two dynamical features that strongly influence the motion and which have previously often been neglected, namely mixing between the particulate flow and the oceanic water and the distribution of sizes suspended by the flow. We show how, in even simple situations, these processes alter our views of how these currents propagate.

Notice that the time is 12:30, not 12:00!

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The following is joint work with Sylvia Serfaty and Cyrill Muratov.

We study the asymptotic behavior of the screened sharp interface

Ohta-Kawasaki energy in dimension 2 using the framework of Γ-convergence.

In that model, two phases appear, and they interact via a nonlocal Coulomb

type energy. We focus on the regime where one of the phases has very small

volume fraction, thus creating ``droplets" of that phase in a sea of the

other phase. We consider perturbations to the critical volume fraction

where droplets first appear, show the number of droplets increases

monotonically with respect to the perturbation factor, and describe their

arrangement in all regimes, whether their number is bounded or unbounded.

When their number is unbounded, the most interesting case we compute the

Γ limit of the `zeroth' order energy and yield averaged information for

almost minimizers, namely that the density of droplets should be uniform.

We then go to the next order, and derive a next order Γ-limit energy,

which is exactly the ``Coulombian renormalized energy W" introduced in the

work of Sandier/Serfaty, and obtained there as a limiting interaction

energy for vortices in Ginzburg-Landau. The derivation is based on their

abstract scheme, that serves to obtain lower bounds for 2-scale energies

and express them through some probabilities on patterns via the

multiparameter ergodic theorem. Without thus appealing at all to the

Euler-Lagrange equation, we establish here for all configurations which

have ``almost minimal energy," the asymptotic roundness and radius of the

droplets as done by Muratov, and the fact that they asymptotically shrink

to points whose arrangement should minimize the renormalized energy W, in

some averaged sense. This leads to expecting to see hexagonal lattices of

droplets.

The task is to estimate approach time (time-to-go (TTG)) of non-ballistic threats (e.g. missiles) using passive infrared imagery captured from a sensor on the target platform (e.g. a helicopter). The threat information available in a frame of data is angular position and signal amplitude.

A Kalman filter approach is presented that is applied to example amplitude data to estimate TTG. Angular information alone is not sufficient to allow analysis of missile guidance dynamics to provide a TTG estimate. Detection of the launch is required as is additional information in the form of a terrain database to determine initial range. Parameters that relate to missile dynamics might include proportional navigation constant and motor thrust. Differences between actual angular position observations and modelled values can beused to form an estimator for the parameter set and thence to the TTG.

The question posed here is, "how can signal amplitude information be employed to establish observability in a state-estimation-based model of the angular data to improve TTG estimate performance without any other source of range information?"

In this part, I will redefine the quantum representations for $G = SU(2)$ making no mention of flat connections at all, instead appealing to a purely combinatorial construction using the knot theory of the Jones polynomial.

Using these, I will discuss some of the properties of the representations, their strengths and their shortcomings. One of their main properties, conjectured by Vladimir Turaev and proved by Jørgen Ellegaard Andersen, is that the collection of the representations forms an infinite-dimensional faithful representation. As it is still an open question whether or not mapping class groups admit faithful finite-dimensional representations, it becomes natural to consider the kernels of the individual representations. Furthermore, I will hopefully discuss Andersen's proof that mapping class groups of closed surfaces do not have Kazhdan's Property (T), which makes essential use of quantum representations.

Ultracold atomic gases have recently proven to be enormously rich

systems from the perspective of a condensed matter physicist. With

the advent of optical lattices, such systems can now realise idealised

model Hamiltonians used to investigate strongly correlated materials.

Conversely, ultracold atomic gases can exhibit quantum phases and

dynamics with no counterpart in the solid state due to their extra

degrees of freedom and unique environments virtually free of

dissipation. In this talk, I will discuss examples of such behaviour

arising from spinor degrees of freedom on which my recent research has

focused. Examples will include bosons with artificially induced

spin-orbit coupling and the non-equilibrium dynamics of spinor

condensates.

Let k be an odd integer and N be a positive integer divisible by 4. Let g be a newform of weight k - 1, level dividing N/2 and trivial character. We give an explicit algorithm for computing the space of cusp forms of weight k/2 that are 'Shimura-equivalent' to g. Applying Waldspurger's theorem to this space allows us to express the critical values of the L-functions of twists of g in terms of the coefficients of modular forms of half-integral weight. Following Tunnell, this often allows us to give a criterion for the n-th twist of an elliptic curve to have positive rank in terms of the number of representations of certain integers by certain ternary quadratic forms.

We will define certain Verdier quotients of the singularity category of a ring R, called defect categories. The triviality of these defect

categories determine, for example, whether a commutative local ring is Gorenstein, or a complete intersection. The dimension (in the sense of Rouquier) of the defect category thus gives a measure of how close such a ring is to being Gorenstein, respectively, a complete intersection. Examples will be given. This is based on joint work with Petter Bergh and Steffen Oppermann.

We study the dispersion and dissipation of the numerical scheme obtained by taking a weighted averaging of the consistent (finite element) mass matrix and lumped (spectral element) mass matrix for the small wave number limit. We find and prove that for the optimum blending the resulting scheme

(a) provides $2p+4$ order accuracy for $p$th order method (two orders more accurate compared with finite and spectral element schemes);

(b) has an absolute accuracy which is $\mathcal{O}(p^{-3})$ and $\mathcal{O}(p^{-2})$ times better than that of the pure finite and spectral element schemes, respectively;

(c) tends to exhibit phase lag.

Moreover, we show that the optimally blended scheme can be efficiently implemented merely by replacing the usual Gaussian quadrature rule used to assemble the mass and stiffness matrices by novel nonstandard quadrature rules which are also derived.

One of the main problems occurring in the aorta is the development of aneurysms, in which case the artery wall thickens and its diameter increases. Suffice to say that many other factors may be involved in this process. These include, amongst others, geometry, non-homogeneous material, anisotropy, growth, remodeling, age, etc. In this talk, we examine the bifurcation of inflated thick-walled cylindrical shells under axial loading and its interpretation in terms of the mechanical response of arterial tissue and the formation and propagation of aneurysms. We will show that this mechanical approach is able to capture features of the mechanisms involved during the formation and propagation of aneurysms.

In this talk I shall discuss about the classical compressible Euler equations in three

space dimensions for a perfect fluid with an arbitrary equation of state.

We considered initial data which outside a sphere coincide with the data corresponding

to a constant state, we established theorems which gave a complete description of the

maximal development. In particular, we showed that the boundary of the domain of the

maximal development has a singular part where the inverse density of the wave fronts

vanishes, signaling shock formation.

In this talk we present a general method using permanent estimates in order to obtain results about counting and packing Hamilton cycles in dense graphs and oriented graphs. As a warm up we prove that every Dirac graph $G$ contains at least $(reg(G)/e)^n$ many distinct Hamilton cycles, where $reg(G)$ is the maximal degree of a spanning regular subgraph of $G$. We continue with strengthening a result of Cuckler by proving that the number of oriented Hamilton cycles in an almost $cn$-regular oriented graph is $(cn/e)^n(1+o(1))^n$, provided that $c$ is greater than $3/8$. Last, we prove that every graph $G$ of minimum degree at least $n/2+\epsilon n$ contains at least $reg_{even}(G)-\epsilon n$ edge-disjoint Hamilton cycles, where $reg_{even}(G)$ is the maximal even degree of a spanning regular subgraph of $G$. This proves an approximate version of a conjecture made by Osthus and K\"uhn.  Joint work with Michael Krivelevich and Benny Sudakov.

I will present the basics of mathematical finance, and what probabilists do there. More specifically, I will present the basic concepts of replication of a derivative contract by trading, market completeness, arbitrage, and the link with Backward Stochastic Differential Equations (BSDEs).

Crystalline solids are descibed by a material manifold endowed

with a certain structure which we call crystalline. This is characterized by

a canonical 1-form, the integral of which on a closed curve in the material manifold

represents, in the continuum limit, the sum of the Burgers vectors of all the dislocation lines

enclosed by the curve. In the case that the dislocation distribution is uniform, the material manifold

becomes a Lie group upon the choice of an identity element. In this talk crystalline solids

with uniform distributions of the two elementary kinds of dislocations, edge and screw dislocations,

shall be considered. These correspond to the two simplest non-Abelian Lie groups, the affine group

and the Heisenberg group respectively. The statics of a crystalline solid are described in terms of a

mapping from the material domain into Euclidean space. The equilibrium configurations correspond

to mappings which minimize a certain energy integral. The static problem is solved in the case of

a small density of dislocations.

It's well known that the mapping space of two finite dimensional

manifolds can be given the structure of an infinite dimensional manifold

modelled on Frechet spaces (provided the source is compact). However, it is

not that the charts on the original manifolds give the charts on the mapping

space: it is a little bit more complicated than that. These complications

become important when one extends this construction, either to spaces more

general than manifolds or to properties other than being locally linear.

In this talk, I shall show how to describe the type of property needed to

transport local properties of a space to local properties of its mapping

space. As an application, we shall show that applying the mapping

construction to a regular map is again regular.